Larry N. Thibos
School of Optometry
Indiana University
Bloomington, IN 47405
thibos
indiana.eduVoice: 812-855-9842
Fax: 812-855-7045
Acknowledgement:
Supported by NEI grant R01-EY05109
Submitted to The CLAO Journal, 19 Aug 02
4 figures, 0 tables
Address for correspondence:
Larry N. Thibos
School of Optometry
Indiana University
Bloomington, IN 47405
thibosindiana.edu
Voice: 812-855-9842
Fax: 812-855-7045
Acknowledgement:
Supported by NEI grant R01-EY05109
Submitted to The CLAO Journal, 19 Aug 02
4 figures, 0 tables
Abstract | Principles of wavefront-guided design | Limitations to correcting aberrations with contact lenses | Conclusion | References
The concept of wavefront-guided design of contact lenses is presented from three vantage points: ray optics, wavefront aberrations, and optical path-length errors. We argue that the goal of contact lenses is to make all of the optical paths from a distant object to the retina equal in length regardless of where the path intersects the plane of the eye’s pupil. The aberration map of an eye is a prescription for such a lens. Unfortunately, variability of measured aberration maps is a fundamental limit to our knowledge of the true aberration structure of an eye. Variability arises because the eye is a biological system that changes over time for normal, physiological reasons. Furthermore, uncertainty in our measurement of the aberration map due to such variable factors as alignment of the aberrometer to the eye by the clinician, or small fixation errors committed by the patient, will make it difficult to achieve a full measure of success with aberration-correcting contact lenses. The clinical implication of these findings is that multiple measurements of the aberration map should be collected using a protocol that includes re-alignment of the instrument and then averaging the aberration maps to reduce the level of uncertainty associated with any single measurement.
The purpose of contact lenses is to correct refractive errors of the eye. From the traditional viewpoint of ray optics, the problem with refractive error is that the eye’s optical system either refracts rays too much, causing them to focus in front of the retina, or the rays are refracted too little, causing them to focus behind the retina. Fig. 1A illustrates this problem of ametropia for a simple case of myopia . The contact lens remedy for this problem, illustrated in Fig. 1B, is to alter the amount of refraction applied to each ray. This is achieved by changing the curvature of the eye’s primary refracting surface, which in the naked eye is the interface between air and the cornea. This curvature change is accomplished by effectively replacing the cornea with the anterior surface of the contact lens. For example, the contact lens for a myopic eye will have an anterior surface which is flatter than the cornea, thereby reducing refracting power so that parallel rays will come to focus on the retina.
An alternative view of refractive error is that the optical wavefronts inside the eye have too much or too little curvature. In a myopic eye, the center of curvature of these wavefronts lies in front of the retina, as shown in Fig. 1A, whereas for a hyperopic eye the center of curvature lies behind the retina. The remedy is to use a contact lens to reshape the wavefront by slowing it down differentially across the pupil. To correct myopic refractive errors, the wavefront must be retarded more at the edges than in the middle of the pupil. Since light travels slower when it is inside the lens (velocity = c/n, where c is speed of light in a vacuum and n is refractive index), the contact lens must be thicker at the edges than in the middle as shown in Fig. 1B.
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Figure 1. Schematic diagrams of the use of a contact lens to correct ametropia.. (A) Ray and wavefront diagram for a myopic eye. (B) A soft contact lens shifts the focus point axially by altering the refraction of rays and by reshaping the wavefront. (C) In an aberration-free myopic eye, the optical path lengths are equal for all rays traveling from the far-point P to its conjugate point P’ on the retina. However, the path lengths from a distance source to P’ are unequal. (D) A contact lens equilibrates path lengths for distant objects by increasing the index of refraction over that portion of the path intercepted by the lens.
A third viewpoint describes the problem of refractive error in terms of optical path lengths. In theory, the optical distance from the eye’s far-point P to the optically conjugate point P' on the retina is the same no matter where a ray from P enters the eye, as shown in Fig. 1C. Recall that optical distance is the product of physical distance times refractive index. This explains why the path PAP´ in this diagram has the same optical length as the path POP´, even though PAP´ is physically longer. In air, the path from P to A is a little longer than the path from P to O. However, inside the eye, where refractive index is greater, the path from A to P´ is physically shorter than the path from O to P´ and is optically much shorter because the physical difference gets multiplied by the refractive index of the eye. Thus, a longer path in air compensates for a shorter path inside the eye. The net result is that the two paths are exactly the same optical length.
Using this language of optical path lengths, the problem of refractive error is that the optical path from a distant point source to the retina is not the same length for every ray entering the eye. In the myopic eye of Fig. 1C, for example, we have seen that the path PAP´ is just the right length, which implies the path BAP´ must be too short. This difference in optical path lengths between the central ray and any other ray from a distant object varies across the pupil and is greatest at the pupil margin. The contact lens remedy is to adjust these optical distances by replacing air, which has refractive index of 1, with the contact lens material, which has a refractive index greater than 1. As shown in Fig. 1D, the thickness of the lens has to vary across the pupil so that all of the optical distances come out equal. In the case of the myopic correction, the lens has to be thicker near the edge to lengthen the optical paths for marginal rays relative to the central rays and thus make all the optical paths the same length.
The concept of optical path length is especially useful for understanding the optical aberrations of contact lenses. For a contact lens material with uniform refractive index, the optical path difference (OPD) is directly proportional to the change in thickness (relative to the pupil center):
where n’ is the index of the lens. For example, if thickness changes as the square of the radius from the pupil center, then the lens will introduce a change in focus without introducing other aberrations. Any other profile of thickness changes will produce aberrations that can be described quantitatively by an aberration map that is directly proportional to the thickness map.
It is worth noting that the direction of light propagation is not indicated in Fig. 1. In fact, these diagrams apply regardless of whether light is propagating from the world to the retina, or from the retina to the world. This is an important observation because objective aberrometers measure the wavefronts of light reflected out of the eye by a point source of light imaged on the retina by the instrument. In this way aberrometers are able to characterize the optical imperfections of eyes despite the fact that the eye is a closed system with an inaccessible image space.
In summary, the wavefront approach to contact lens design is to flatten the wavefront that emerges from the eye produced by a point source placed on the retina. Conceptually, this is equivalent to making all of the optical paths from a distant object to the retina equal regardless of where the paths enter the eye’s pupil. In the simple case of spherical refractive errors (i.e. myopia and hyperopia), the amount of correction required is the same for all meridia. Thus we need to specify only one number, the power of the contact lens. For astigmatic corrections, every meridian is different but the changes between meridia are systematic so it is sufficient for a prescription to specify the powers of the two orthogonal, principle meridia and their orientation angle. But in general, eyes have higher-order aberrations in addition to the second-order aberrations of sphere and cylinder, which means that the correction required is different for every point in the pupil and cannot be described by just one or three numbers. Thus a “prescription for perfection”, in the general case, requires a two-dimensional map of wavefront errors, or equivalently, optical path differences for every point in the pupil. For simplicity, we call this an “aberration map”.
In theory, it would seem straightforward to measure the aberration map and then use that map as a specification for cutting a contact lens to correct all of the eye’s aberrations, thereby making the eye optically perfect. The 2-dimensional thickness profile for the correcting lens would be obtained by scaling the eye’s OPD map by the factor 1/(1-n’) in accordance with equation (1). In practice, however, there are at least three limitations to a successful outcome.
First, to realize the gains in optical quality resulting from the correction of higher-order aberrations requires that the lower-order aberrations of sphere and cylinder be reduced to much smaller levels than are currently acceptable. In a recent study of well-corrected eyes, we found that the total amount of higher-order aberrations was equivalent to about 0.25 D. 1 This means that the contact lens must correct sphero-cylindrical errors to a small fraction of this value or else the gains from correcting the higher order aberrations could be substantially reduced and potentially even reversed. Achieving accurate focus for polychromatic light will be especially challenging in the presence of ocular chromatic aberration.
Second, the alignment requirements for correcting higher-order aberrations are closer to the stringent requirements for correcting astigmatism with toric lenses than to the relatively slack requirements for correcting myopia with spherical lenses. It is well known that the effectiveness of a standard toric lens requires that it be rotationally stable. In the same way, to eliminate higher-order aberrations requires that the lens be well-aligned with the eye. Any movement of the lens due to blinking or eye rotation will diminish the effectiveness of the correcting lens. As a lens seeks to correct higher and higher order aberrations, rotational and positional alignment must become more and more accurate and thus the lens must be more stable. Recent quantitative analysis of this problem suggests that gains in optical quality are possible, but achieving perfect optical quality is unlikely given the level of stability achieved by current lens designs. 2, 3
Figure 2 Examples of variability in aberration maps measured on three different time scales using a clinical aberrometer (COAS, by Wavefront Sciences, Inc.). Each map shows the optical path length errors (relative to the pupil center) across a 6mm diameter pupil, displayed as a contour map (0.1 µm intervals). First-order (prism) and second-order (sphere and cylinder) aberrations have been subtracted from the maps to emphasize higher order (3rd and 4th order) aberrations.
The third limitation is that we have incomplete knowledge of the true aberration map of an eye for at least two reasons. First, the eye is a biological system that changes over time for normal, physiological reasons. Thus the aberration map measured today may not be appropriate tomorrow. Changes in the aberration map with accommodation are another source of biological variability. 4 Furthermore, uncertainty in measurement of the aberration map due to such variable factors as alignment of the aberrometer to the eye by the clinician, or small fixation errors committed by the patient, will make it difficult to achieve a full optical correction with a contact lens.
To illustrate the problem of temporal variability of the aberration map, we show in Fig. 2 pairs of aberration maps obtained over different time spans from a single individual. To concentrate our attention on the higher-order aberrations, the lower order aberrations have been mathematically removed from the map. The upper pair of maps was obtained within one second of each other while the patient was in a freeze position where he didn’t move, blink, or breathe. Notice that the maps are nearly identical, which indicates that under these conditions the aberration map is extremely stable. The middle pair of maps was obtained with an interval of a few minutes. Between measurements the patient was allowed to blink and move about, with the clinician re-aligning the aberrometer for each measurement. Notice that the maps are still very similar to each other, but subtle differences are noticeable upon close inspection. The lower pair of maps was obtained at about the same time of day, but on different days of the same week. Now it is possible to see large differences in the aberration maps.
Figure 3. Analysis of variability in aberration maps. (A) Mean across 4 subjects of the standard deviation in RMS wavefront error measured on different time scales. Analysis included third and fourth-order aberrations only. (B) Standard deviation of RMS errors measured on a model eye in a single session and for multiple sessions in which the aberrometer was re-aligned for each measurement.
We repeated this experiment five times on four patients and the results are summarized quantitatively in Fig. 3A. The magnitude of the higher-order aberrations is quantified by a measure called the RMS error, which describes how warped the aberration map is. The standard deviation of N=5 repeated measures of the RMS error was computed for each patient and these results were then averaged across patients to get a sense of the average amount of variability. When the five measurements were collected over a period of less than 1 second, the average RMS error was 0.01 µm as shown by the upper bar in Fig. 3A. Variability almost doubled when the data were collected over a period of several minutes, as shown in the middle bar of Fig. 3A, and grew even more when collected over 1 week.
To judge the clinical significance of these results, it is helpful to convert from microns of RMS error to diopters of equivalent defocus. Equivalent defocus is defined as the amount of ordinary defocus needed to produce the same RMS wavefront error that is produced by one or more higher-order aberrations. It is computed using the formula
It is important to bear in mind that 1 diopter of ordinary defocus doesn’t necessarily have the same effect as 1 diopter of equivalent defocus because different types of aberrations affect the retinal image in different ways. Nevertheless, by expressing RMS error in terms of equivalent defocus the data are put into familiar units that help us judge the order of magnitude of the effect. From the diopter scale in Fig. 3 it is clear that the levels of variability of higher order aberrations encountered in our experiments were much smaller than the variability of clinical refraction. 5 This is understandable because, as a rule, the variability of large numbers is large and the variability of small numbers is small. Thus we expect the variability of the relatively weak, higher order aberrations to be much smaller than the variability of spherical refractive errors.
Our discovery that the variability of higher-order RMS error grows with the time scale over which measurements are taken leads us to ask, what is the source of this variability? To approach this question we repeated the experiment using a physical model eye with about the same amount of higher-order RMS error as the typical human eye. These results are shown in Fig. 3B. We found that if multiple measurements were taken during a single session, where the alignment between the aberrometer and model eye was fixed, then the degree of variability was extremely small (0.0015 D equivalent defocus). However, when the aberrometer was re-aligned between measurements of the model eye, variability increased by a factor of 10 to about the same level as we had measured for human eyes under similar conditions. This suggests that the variability we encountered on human eyes when taking measurements over the course of several minutes could be accounted for by re-alignment of the instrument, rather than biological variability. However, the small increase in variability observed on a weekly time scale probably reflects real, biological changes. The clinical implication of these findings is that multiple measurements of the aberration map should be collected using a protocol that includes re-alignment of the instrument and then averaging the aberration maps to reduce the level of uncertainty associated with any single measurement.
Figure 4. Effect of variability in the aberration map on monochromatic image quality of the eye. The modulation transfer function was computed for diffraction-limited optics (upper curve), for the case of sphero-cylindrical correction of the eye’s aberrations (lower curves), and for the case where the aberration map measured one day is used to correct the higher-order aberrations measured on a different day.
The question remains, what is the likely impact of this variability on vision? Although higher-order aberrations are relatively small, they can make a big difference in image quality if the eye’s sphero-cylindrical errors are fully corrected. An example is shown in Fig. 4. If the eye of one of our patients is perfectly corrected only for sphere and cylinder, the modulation transfer function (MTF) is significantly worse than the MTF for an optically perfect eye. Raising this performance to reach perfection is the aim of wavefront-guided contact lens design. However, if we use today’s prescription to correct tomorrow’s eye, the aberration map would have changed enough to significantly limit the gain expected from the ideal contact lens.
Realization of the potential gains in image quality and vision by correcting higher order aberrations with contact lenses will be limited by several factors. The current study emphasizes that measurement variability and biological variability may both play a role in reducing the benefits of wavefront correction. We specifically recommend, therefore, that efforts to minimize the impact of measurement variability (e.g. by averaging across a number of individual measurements) be included in clinical aberrometry.
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